String Theory Demystified

CHAPTER 5 Conformal Field Theory Part I
µ µ
Using =∂µ ∂ , taking the derivative with respect to ∂ we have, on the left-hand
side:
µ
µ
∂ ∂ ε +∂ ∂ ε = ε +∂ ( ∂⋅ε)
µ ν
ν µ ν ν
Hence, taking the same derivative on the right side and equating results we obtain
⎛ ⎞
εν + − ν ε
⎝
⎜1
⎠
⎟ ∂ ∂⋅ =
2
( ) 0
d
Notice that this equation singles out the case of two dimensions. Setting d = 2
we obtain
ε ν = 0
We can obtain a second equation which highlights the importance of d = 2 by
µ
operating on * with =∂∂ µ . This gives
{ δ + ( d −2) ∂ ∂ }( ∂⋅ ε)
= 0
µν µ ν
The infi nitesimal parameter ε µ can represent four different types of transformations:
translations, scale transformations, rotations, and special conformal transformations.
A translation takes the form
µ µ
ε = a
where a µ is a constant. A scale transformation is one of the form:
For a rotation, we write
µ µ
ε = λx
µ µ
ε = ω x
where we require that ω is antisymmetric, that is, ωµν =− ωνµ
. Finally, a special
conformal transformation assumes the form
ε
= b x −2 x ( b⋅x) ν
ν
µ µ 2 µ
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